Following Fuglede [1]}, a kernel \(\kappa \)  on \(X\)  is meant to be a symmetric, l.s.c. function \(\kappa :X\times X\rightarrow (-\infty ,\infty ]\)  that is positive (\({}\geqslant 0\) ) unless the space \(X\)  is compact.The latter case can often be reduced to the former by introducing a kernel \(\kappa ^{\prime }:=\kappa +q\geqslant 0\) , where \(q\in \mathbb {R}\) , which is always possible, for a l.s.c. function on a compact space is lower bounded.
Given (signed) measures \(\mu ,\nu \in \mathfrak {M}\) , define the potential and the mutual energy by
\(\kappa \mu (x)&:=\int \kappa (x,y)\,d\mu (y),\quad x\in X,\\\kappa (\mu ,\nu )&:=\int \kappa (x,y)\,d(\mu \otimes \nu )(x,y),\) 
